A Generalized Frequency Response for Nonlinear Systems

The principal aim of this paper is to present the idea of ‘generalizedfrequency response’ of a nonlinear input-output map S_xo.It is defined as · S_xo · ^-1 where is the canonicalisomorphsim from the weighted space L²_w[0, ] to ² with respectto a given basis for L²_w(0 , ) and the standard basis for ².Realization results are presented pertainig to the linear, bilinear,and nonlinear cases.     

A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following thermoelasticplate equation with Dirichlet boundary conditions w_tt + ²w + = a₁(x)u₁ + ... + a_m(x)u_m, t 0, x , _t – ß – w_t = d₁(x)u₁ + ... + d_m(x)u_m,t 0, x , = w = w = 0, t 0, x , where 0, ß > 0, is a sufficiently regular boundeddomain in R^N, a_i, d_i, L² (; R), the control functions u_i L²(0, t₁; R); i = 1, 2, ..., m. This condition is easy to checkand is given by Rank [P_jBA_jP_jBA²_jP_jB ... A^3_j–1_jP_jB] = 3_j,BU=b₁U₁+...+b_mU_m,b_i=[0, a_i, d_i], A_j=[0, –²_j, 0, 1, 0, –_j, 0, _j, –ß_j]P_j, j1, where _j, S are the eigenvalues of – with Dirichlet boundarycondition and P_j, S are the projections on the correspondingeigenspace.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

Approximation of exact controllability problem involving parabolic differential equations

In this paper we examine computation of optimal control u^* ofthe exact controllability problem (referred to as the constraintproblem) governed by the following type of linear parabolicdifferential equations: (y/t) + Ay = u in Q y = 0 on y(0) = y₀ on where A is the second-order elliptic differential operator, is a bounded domain in ^k with smooth boundary , Q = (0, T)x , = (0, T) x and T > 0. This is achieved by approximatingu^* through a sequence {u_n} of controls corresponding to unconstrainedproblems involving a penalty function arising from the controllabilityconstraint.     

Total Flux Estimates for a Finite-Element Approximation of Parabolic Equations

An initial-boundary-value problem for a parabolic equation ina domain x (0, T) with prescribed Dirichlet data on is approximatedusing a continuous-time Galerkin finite-element scheme. It isshown that the total flux across ₁= can be approximated withan error of O(h^k) when is a curved domain in Rⁿ (n = 2 or 3)and isoparametric elements having approximation power h^k inthe L² norm are used.     

A generalization of the concept of similarity of infinite-dimensional linear systems with applications to stability, stabilizability, and detectability

In the present paper a generalization of the concept of similaritycalled D similarity for infinite-demensional linear systemsis introduced. We show that this D similarity preserves thespectrum in the standard sense, and the exponential growth bound,stability, stabilizability, and detectability in the extensionsense. For the case where one of the semigroups is stronglydifferentiable at some t₀ 0, it is shown that D similaritypreserves the exponential growth bound and weak and exponentialstability in the standard sense. For the case where one of thesemigroups is analytic, it is shown that D similarity preservesstabilizability and detectability in the standard sense.     

Hankel operators of systems with feedthrough delays

We study the singular values of and the essential spectrumof * when is the (non compact) Hankel operator of a systemwith feedthrough delays.     

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u² [(v)]) - (1)/(3) .(u³ w)= 0, w = - c u - u^-+ a u^-3 , (v)/(t) + . (u v [(v)]) - v - .(u² v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented.     

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following systemof parabolic equations with Dirichlet boundary condition: {z_t = D z + b₁(x)u₁ + ··· + b_m(x)u_m, t 0, z ⁿ, z = 0, on where is a sufficiently smooth bounded domain in ^N, b_i L²(;ⁿ), the control functions u_i L²(0, t₁; ); i = 1, 2, ..., mand D is an n x n non-diagonal matrix whose eigenvalues aresemi-simple with positive real part. This algebraic conditionis checkable since it is given in terms of the n_j x m matricesDP_j and P_jB, i.e. Rank [P_jBDP_jBD²P_jB··· D^n_j–1 P_jB]= n_j, where P_jBu = P_jb₁u₁ + ··· + P_jb_mu_m. Finally,this result can be applied to those systems of partial differentialequations that can be rewritten as a diffusion system (see deOliveira, 1998).     

Robust H{infty} dynamic output feedback control for 2D linear parameter-varying systems

James Lam Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Changhong Wang Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, People's Republic of China Corresponding author. Email: ligangwu{at}hit.edu.cn Email: james.lam{at}hku.hk Email: cwang{at}hit.edu.cn Received on June 28, 2006; Accepted on October 1, 2007 This paper is concerned with the problem of robust dynamicoutput feedback control for 2D discrete-time linear parameter-varyingsystems. Given a Fornasini–Marchesini local state-spacesystem with linear varying parameters, our attention is focussedon the design of full-order dynamic output feedback controller,which guarantees the closed-loop system to be asymptoticallystable and has a prescribed disturbance attenuation performance.A sufficient condition for the existence of a desired robustoutput feedback controller is established in terms of parameterizedlinear matrix inequalities, and the corresponding controllersynthesis is cast into a convex optimization problem which canbe efficiently handled by using standard numerical software.A numerical example is provided to illustrate the effectivenessof the proposed design method.     

Characterization of Distributions through Maximization of {alpha}-Entropies

This paper provides characterizations of the Pareto distributionof the second kind, the Pearson type-VII distribution and thet-distribution. These characterizations have been obtained throughmaximization of -entropies. As 1, characterizations of the exponential,the Laplace, and the normal distributions are obtained. On sabbatical leave from the Department of Statistics, YarmoukUniversity, Irbid, Jordan.     

Theta-characteristics on singular curves

On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh⁰(C, ) 0 and h⁰(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).     

A Note on a System Theoretic Approach to a Conjecture by Peller-Khrushchev: The General Case

Based on the construction of infinite-dimensional balanced realizations,an alternative solution to the following inverse spectral problemis presented. Given a decreasing sequence of positive numbers(_n)_n>1 (i.e. ₁₂₃...0), does there exist a Hankel operatorwhose sequance of singular values is (_n)_n>1? This paper isan extension of a previously published paper in which the sameapproach was taken in the case of a monotonically decreasingsequence(_n)_n>1.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

On the Optimum Criterion of Polynomial Stability

The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a₀xⁿ+a₁x^n–1+...+a_n be apositive-coefficient polynomial. The numbers ₁=a_i-1a_i+2/a_ia_i+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif _i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)²=1.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

Blow up of solutions of a generalized Boussinesq equation

Consider the Cauchy problem u_tt = (f(u))_xx + u_xxtt x R, t 0, u(x,0) = u₀(x), u_t(x,0) = u₁(x),7rcub; where f : R R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let > 0 be real, such that 2(l + 2)F(u) uf(u), (v₀, Pv₀)_l² + _- F(u₀)dx < 0 where P = 1 - d²/dx², F'(s), and v₀ is given by u₁(x,0) = (v₀(x))_x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.     

Compact Embeddings of Besov Spaces in Exponential Orlicz Spaces

Let 1 < p < , 0 < v < p', let be a bounded domainin Rⁿ, and denote by id the limiting compact embedding of theBesov space (Rⁿ) into the exponentialOrlicz space L_exp(t^v)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.